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Let X be a continuous random variable with mean My and variance of and let a and be any constant fixed numbers. Define the random variable Was follows: W= a + bx Then the mean and variance of Ware w = [+bX] = a + bux (5.71 and c = Vara + hX ] = bar (5.8) and the standard deviation of Wis (5.9) An important special case of these results is the standardized random variable 46 X - MX 2 = (5.10) TX which has mean 0 and variance 1. Linear functions of random variables have many applications in business and eco- nomics. Suppose that the number of units sold during a week is a random variable and the selling price is fixed. Thus, the total revenue is a random variable that is a function of the random variable units sold. Quantity demanded is a linear function of price that can be a random variable. Thus, quantity demanded is a random variable. The total number of cars sold per month in a dealership is a linear function of the random variable number of cars sold per sales person multiplied by the number of sales persons. Thus, total sales is a random variable Example 5.2 Home Heating Costs (Mean and Standard Deviation) A homeowner estimates that within the range of likely temperatures his January heat- ing bill, Y, in dollars, will be Y = 290 - 57 where Tis the average temperature for the month, in degrees Fahrenheit. If the average January temperature can be represented by a random variable with a mean of 24 and a standard deviation of 4, find the mean and standard deviation of this homeowner's January heating bill